what Prefered Numbers are

The number system in common use is base 10, so we likely want to have 10, 100, 1000 in there. That's not fine-grained enough for practical use in most cases. Let's make it evenly spaced in between each magnitude and explore for seven and three steps, each.

› $a = 10
10
› sprintf '%.f', $a += (100-10)/7
23
› sprintf '%.f', $a += (100-10)/7
36
› sprintf '%.f', $a += (100-10)/7
49
› sprintf '%.f', $a += (100-10)/7
61
› sprintf '%.f', $a += (100-10)/7
74
› sprintf '%.f', $a += (100-10)/7
87
› sprintf '%.f', $a += (100-10)/7
100

› $a = 10
10
› sprintf '%.f', $a += (100-10)/3
40
› sprintf '%.f', $a += (100-10)/3
70
› sprintf '%.f', $a += (100-10)/3
100
[download]

› $a = 10
10
› sprintf '%.f', $a *= 10**(1/7)
14
› sprintf '%.f', $a *= 10**(1/7)
19
› sprintf '%.f', $a *= 10**(1/7)
27
› sprintf '%.f', $a *= 10**(1/7)
37
› sprintf '%.f', $a *= 10**(1/7)
52
› sprintf '%.f', $a *= 10**(1/7)
72
› sprintf '%.f', $a *= 10**(1/7)
100

› $a = 10
10
› sprintf '%.f', $a *= 10**(1/3)
22
› sprintf '%.f', $a *= 10**(1/3)
46
› sprintf '%.f', $a *= 10**(1/3)
100
[download]

# playground
my ($start, $end) = (10, 100);  # try: (35, 7400)
for my $step (1..20) {
    printf 'step %2d:', $step;
    my $i = $start;
    for (1..$step+1) {
        printf ' %.f', $i;
        $i *= ($end/$start)**(1/$step);
    }
    print "\n";
}
__END__
step  1: 10 100
step  2: 10 32 100
step  3: 10 22 46 100
step  4: 10 18 32 56 100
step  5: 10 16 25 40 63 100
step  6: 10 15 22 32 46 68 100
step  7: 10 14 19 27 37 52 72 100
step  8: 10 13 18 24 32 42 56 75 100
step  9: 10 13 17 22 28 36 46 60 77 100
step 10: 10 13 16 20 25 32 40 50 63 79 100
step 11: 10 12 15 19 23 28 35 43 53 66 81 100
step 12: 10 12 15 18 22 26 32 38 46 56 68 83 100
step 13: 10 12 14 17 20 24 29 35 41 49 59 70 84 100
step 14: 10 12 14 16 19 23 27 32 37 44 52 61 72 85 100
step 15: 10 12 14 16 18 22 25 29 34 40 46 54 63 74 86 100
step 16: 10 12 13 15 18 21 24 27 32 37 42 49 56 65 75 87 100
step 17: 10 11 13 15 17 20 23 26 30 34 39 44 51 58 67 76 87 100
step 18: 10 11 13 15 17 19 22 24 28 32 36 41 46 53 60 68 77 88 100
step 19: 10 11 13 14 16 18 21 23 26 30 34 38 43 48 55 62 70 78 89 100
step 20: 10 11 13 14 16 18 20 22 25 28 32 35 40 45 50 56 63 71 79 89 1
+00
[download]

that's super thanks.

Your last comment about money denominations got me thinking: another aspect of these numbers is to be able to construct sums with the shortest linear combination - e.g. use the fewest different money denominations for change of 39.43. So that's an added optimisation parameter right? Is there theory for that?

Is there theory for that?

Isn't it as simple as prime factors of the base (with the additional 1, of course)?

Edit: not necessarily prime, d'oh!

A familiar (if you're a wirehead) example of preferred series are the standardized resistor values for various tolerances.

i get it!

I know you've said you better understand it now... but there was a small point that I like, that I don't think was brought out enough. Like AnomalousMonk's link to resistor tolerances implies (but doesn't say explicitly enough, IMO): one of the nice things about the series is that it keeps allowable tolerance consistent. If you're manufacturing widgets, and your manufacturing has 0.1mm tolerance, then doing widgets in sizes 1,2,3,4,5,6,7,8,9,10 is reasonable; none of the widget sizes will overlap. But if you manufacture with 10% tolerance, then the step from 1 to 2 is fine, because 1+10%=1.1 doesn't overlap with 2-10%=1.8. But at the step from 9 to 10, 9+10%=9.9 and 10-10%=9, so a value of 9 is a technically valid 9+-10% or 10+/-10% widget. If you count the "maximum allowable tolerance" as the point where the lower value shifted up by the tolerance and the upper value shifted down by the tolerance are equals, then you can see:

#!/usr/bin/env perl
# Preferred Numbers: https://perlmonks.org/?node_id=11103336

my $prev = 1;
for my $this ( 2 .. 10 ) {
    # an x% change can fit between $prev and $this
    # x% changes, depending on $this
    # Overlap will start if x% is such that prev * (1+x%) == this * (1
+-x%)
    #   algebra:
    #       p*(1+x) = t*(1-x)
    #       p + px = t - tx
    #       (p+t)x = t-p
    #       x = (t-p)/(p+t)
    #       x% = 100%*x
    my $xpct = ($this-$prev) / ($this+$prev) * 100;
    printf "linear: %.2f .. %.2f => %.2f%%\n", $prev, $this, $xpct;
} continue {
    $prev = $this;
}

print "\n";

my $prev = 1;
for my $i ( 1 .. 10 ) {
    # now go in 10 logarithmic steps
    $this = $prev * (10**(1/10));
    my $xpct = ($this-$prev) / ($this+$prev) * 100;
    printf "logarithmic: %.2f .. %.2f => %.2f%%\n", $prev, $this, $xpc
+t;
} continue {
    $prev = $this;
}

print "\n";

my $prev = 1;
for my $i ( 1 .. 10 ) {
    # now go in 10 logarithmic steps
    $this = 10**($i/10);
    my $round_p = int($prev*10+0.5)/10;
    my $round_t = int($this*10+0.5)/10;
    my $xpct = ($round_t-$round_p) / ($round_t+$round_p) * 100;
    printf "rounded logarithmic: %.2f .. %.2f => %.2f%%\n", $round_p, 
+$round_t, $xpct;
} continue {
    $prev = $this;
}

__END__
linear: 1.00 .. 2.00 => 33.33%
linear: 2.00 .. 3.00 => 20.00%
linear: 3.00 .. 4.00 => 14.29%
linear: 4.00 .. 5.00 => 11.11%
linear: 5.00 .. 6.00 => 9.09%
linear: 6.00 .. 7.00 => 7.69%
linear: 7.00 .. 8.00 => 6.67%
linear: 8.00 .. 9.00 => 5.88%
linear: 9.00 .. 10.00 => 5.26%

logarithmic: 1.00 .. 1.26 => 11.46%
logarithmic: 1.26 .. 1.58 => 11.46%
logarithmic: 1.58 .. 2.00 => 11.46%
logarithmic: 2.00 .. 2.51 => 11.46%
logarithmic: 2.51 .. 3.16 => 11.46%
logarithmic: 3.16 .. 3.98 => 11.46%
logarithmic: 3.98 .. 5.01 => 11.46%
logarithmic: 5.01 .. 6.31 => 11.46%
logarithmic: 6.31 .. 7.94 => 11.46%
logarithmic: 7.94 .. 10.00 => 11.46%

rounded logarithmic: 1.00 .. 1.30 => 13.04%
rounded logarithmic: 1.30 .. 1.60 => 10.34%
rounded logarithmic: 1.60 .. 2.00 => 11.11%
rounded logarithmic: 2.00 .. 2.50 => 11.11%
rounded logarithmic: 2.50 .. 3.20 => 12.28%
rounded logarithmic: 3.20 .. 4.00 => 11.11%
rounded logarithmic: 4.00 .. 5.00 => 11.11%
rounded logarithmic: 5.00 .. 6.30 => 11.50%
rounded logarithmic: 6.30 .. 7.90 => 11.27%
rounded logarithmic: 7.90 .. 10.00 => 11.73%
[download]

The linear scale wastes a lot of the tolerance near the low end, so you can have 30%, but at the high end, you can really only have 5% tolerance on the widgets. However, if you're doing 5% widgets, at the low end, you'll have big gaps, and a customer will complain that they want an intermediate value. With the even-logarithmic steps, every step between sizes will have the same amount of "room" for the tolerances, so if you can manufacture the widgets with 10% tolerance, they'll all nicely fit, with no overlap between sizes. (Even with some rounding on the Preferred Numbers in the 10-logarithmic steps will still show about the same tolerance range, and can handle 10% tolerance just fine.)

I am interested in manufacturing, from the point of centrally planning an economy. Albeit, I had no practical exposure to it ever and so all this stuff and your explanation is interesting and helpful.

Wouldn't increasing tolerances just for the sake of not overlapping, going to be a headache for users? Is this "best practice" in design? I mean I buy 2 screws and the bigger one has double (percentage-wise) tolerance than that of the smaller one!

... increasing tolerances just for the sake of not overlapping ... 2 screws and the bigger one has double (percentage-wise) tolerance ...

But the tolerances of the items in question do not change, only their ranges of absolute value. E.g., for 10% (i.e., +/-10%) resistors, a 100 ohm resistor has a range of 90 to 110 ohms, the 82 ohm resistor "below" it ranges over 73.8 - 90.2 ohms, and the 120 ohm resistor "above" it ranges over 108 - 132 ohms. At the other end of that magnitude, 820, 1000 and 1200 ohm 10% resistors range over 738 - 902, 900 - 1100 and 1080 - 1320 ohms respectively.

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